Combine the first 10 ingredients in a blender or food processor and puree until smooth, stopping to scrape down the sides as needed. Place the chicken pieces in a baking dish or zip-top plastic bag and pour the marinade over the chicken. Marinate for 2 hours, or up 4 hours, turning occasionally to coat.
Preheat the grill to medium.
Remove meat from marinade, discarding the marinade, and pat the chicken dry with paper towels. Lightly season the chicken pieces with additional salt and pepper on both sides. Place the chicken on the grill and cook for 8 to 10 minutes on the first side. Turn and cook an additional 8 to 10 minutes. Turn chicken, as needed to keep from burning and to get grill marks until chicken is cooked through and tender, about 25 to 30 minutes total for thighs and legs and 35 to 40 minutes for chicken leg quarters. (An instant-read thermometer inserted into the thickest part of the chicken should register 165 degrees F, when fully cooked through.)
Remove chicken from the grill and let stand for 5 minutes before serving. Garnish chicken with fresh cilantro sprigs and lime wedges before serving.